发表机构
Steklov Mathematical Institute（斯捷克洛夫数学研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文利用完全图上的尖锐log-Sobolev不等式，为von Neumann熵和形成纠缠度建立了尖锐的Lipschitz下半连续界，并证明最优常数与log-Sobolev常数一致。
AI 中文摘要
我们给出了由完全图上的均匀测度的尖锐log-Sobolev不等式推导出的两个结果，这些结果涉及有限维和无限维量子系统状态的von Neumann熵和形成纠缠度。第一个结果是von Neumann熵在任意具有均匀正谱（即与投影算子成比例的状态）的混合态$\rho$处关于保真度亏损的尖锐Lipschitz下半连续界：不等式$\\,S(\rho)-S(\sigma)\leq C_\rho(1-F(\rho,\sigma))\\,$对任意状态$\sigma$成立，其中$C_{\rho}$是依赖于$\rho$的秩的常数。第二个结果是形成纠缠度在任意具有均匀正谱边缘态的纯态$\rho$处的尖锐Lipschitz下半连续界：不等式$\\,E_F(\rho)-E_F(\sigma)\leq \frac{1}{2}\\,C_\rho\\|\rho-\sigma\\|_1\\,$对任意状态$\sigma$成立，其中$C_{\rho}$是依赖于$\rho$的Schmidt秩的常数。在这两种情况下，最优常数$C_\rho$等于具有$d$个顶点的完全图的log-Sobolev不等式中的最优常数$K_{d}$：在第一种情况下$d=\mathrm{rank}\rho$，在第二种情况下$d=\mathrm{rank}\rho_A=\mathrm{rank}\rho_B$。作者感谢GPT 5.6在准备本注释过程中提供的宝贵讨论和技术帮助。
英文摘要
We present two results derived from the sharp log-Sobolev inequality for the uniform measure on a complete graph which concern the von Neumann entropy and the Entanglement of Formation of a state of finite and infinite-dimensional quantum systems. The first result is a sharp Lipschitz lower semicontinuity bound for the von Neumann entropy at any mixed state $ρ$ with uniform positive spectrum (i.e. a state proportional to a projector) w.r.t. the fidelity deficit: the inequality $\,S(ρ)-S(σ)\leq C_ρ(1-F(ρ,σ))\,$ valid for any state $σ$, where $C_ρ$ is a constant depending on the rank of $ρ$. The second result is a sharp Lipschitz lower semicontinuity bound for the Entanglement of Formation at any pure state $ρ$ with uniform positive spectrum of marginal states w.r.t. the fidelity deficit: the inequality $\,E_F(ρ)-E_F(σ)\leq C_ρ(1-\mathrm{Tr}ρσ)\,$ valid for any state $σ$, where $C_ρ$ is a constant depending on the Schmidt rank of $ρ$. In both cases the optimal constant $C_ρ$ is equal to the optimal constant $K_{d}$ in the log-Sobolev inequality for the complete graph with $d$ vertices: in the first case $d=\mathrm{rank}ρ$, in the second one $d=\mathrm{rank}ρ_A=\mathrm{rank}ρ_B$. The authors are grateful to GPT 5.6 for valuable discussion and technical help in preparing this note.
Comments14 pages, any comments are welcome
